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# Paper: Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

## Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

Costas-Santos, R. S. and Soria-Lorente, A. Journal of Difference Equations and Applications 24, no. 11 (2018), 1715—1733

MDPI (SWITZERLAND) | ISSN: 2073-8994 | JCR® 2018 Impact Factor: 2.143 - MULTIDISCIPLINARY SCIENCES — position: 30/69 (Q2/T2)

## Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product $$\langle f, g\rangle_S :=\langle {\bf u}, f\,g\rangle+N({\mathcal D}_q\, f)(\alpha)({\mathcal D}_q\, g)(\alpha)$$, where $$\alpha\in \mathbb R, N\ge 0$$, and $${\bf u}$$ is a $$q$$-classical linear functional and $${\mathcal D}_q$$ is the $$q$$-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear $$q$$-difference holonomic equation fulfilled by such polynomials.

We present an analysis of the behaviour of its zeros as a function of the mass $$N$$. In particular, we obtain the exact values of $$N$$ such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work considering two examples.

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## BibTeX

@article {MR3883341,
AUTHOR = {Costas-Santos, R. S. and Soria-Lorente, Anier},
TITLE = {Analytic properties of some basic hypergeometric-{S}obolev-type orthogonal polynomials},
JOURNAL = {J. Difference Equ. Appl.},
FJOURNAL = {Journal of Difference Equations and Applications},
VOLUME = {24},
YEAR = {2018},
NUMBER = {11},
PAGES = {1715--1733},
ISSN = {1023-6198},
MRCLASS = {33D45 (05A30 39A13)},
MRNUMBER = {3883341},
ZBL = {3A1405.33024},
DOI = {10.1080/10236198.2018.1517760},
URL = {https://doi.org/10.1080/10236198.2018.1517760},
}